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The one-dimensional wave equation with general boundary conditions

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Abstract

We show that a realization of the Laplace operator Au := u′′ with general nonlocal Robin boundary conditions α j u′(j) + β j u(j) + γ 1–j u(1 − j) = 0, (j = 0, 1) generates a cosine family on L p(0, 1) for every \({p\,{\in}\,[1,\infty)}\). Here α j , β j and γ j are complex numbers satisfying α 0, α 1 ≠ 0. We also obtain an explicit representation of local solutions to the associated wave equation by using the classical d’Alembert’s formula.

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References

  1. Arendt W. et al.: Vector-valued Laplace Transforms and Cauchy Problems. Birkhäuser, Basel (2001)

    MATH  Google Scholar 

  2. Bobrowski A.: Generation of cosine families via Lord Kelvin’s method of images. J. Evol. Equ. 10, 663–675 (2010)

    Article  MathSciNet  Google Scholar 

  3. R. Chill, V. Keyantuo, and M. Warma, Generation of cosine families on L p(0, 1) by elliptic operators with Robin boundary conditions, Functional analysis and evolution equations, 113–130, Birkhäuser, Basel, 2008.

  4. Fattorini H.O.: Second Order Linear Differential Equations in Banach Spaces. North-Holland Publishing Co., Amsterdam (1985)

    MATH  Google Scholar 

  5. Goldstein J.A.: Semigroups of Linear Operators and Applications. Oxford University Press, New York (1985)

    MATH  Google Scholar 

  6. Kisyński J.: On cosine operator functions and one-parameter groups of operators. Studia Math. 44, 93–105 (1972)

    MathSciNet  Google Scholar 

  7. Xiao T.-J., Liang J.: Second order differential operators with Feller-Wentzell type boundary conditions. J. Funct. Anal. 254, 1467–1486 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Mahamadi Warma.

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Alvarez-Pardo, E., Warma, M. The one-dimensional wave equation with general boundary conditions. Arch. Math. 96, 177–186 (2011). https://doi.org/10.1007/s00013-010-0209-y

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  • DOI: https://doi.org/10.1007/s00013-010-0209-y

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